Determination of the Parameters of Ogden's Law Using Biaxial Data and Levenberg-Marquardt- Fletcher Algorithm

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Abstract
Ogden's constitutive law is often used in finite element simula tions of the behavior of mechanical parts containing rubber. This law has a phenomenological foundation and involves a serie of material parameters μ., α, which must be experimentally determined. Most of the works carried out in this field were based on experimental data published by Treloar in 1944. Theses results concern simple types of strain: simple tension, pure shear, or equi-biaxial tension. This single parameter validation is insufficient. Besides, the three one-parameter validations lead to different sets of material constants. In the present work we use two-parameter experimental data given by a biax ial extension experimental technique: a cylindrical tubular sample with a thin wall is either stretched along its major axis or inflated uniformly. This tech nique allows the imposition of deformation along the two principal directions in an independent manner (1 ≤ λ, ≤ 2.7; 1 ≤ λ 2 ≤ 2.5). The third stretch is deduced by using the incompressibility property. The material constants are then numerically determined by means of a syste matic optimization procedure based on the adaptation of the Levenberg- Marquardt-Fletcher algorithm, initiated by Levenberg-Marquardt (1963) and recently modified by Fletcher (1987). Ogden's law parameters numerically determined by this procedure permit one to calculate numerical values of the principal stresses corresponding to the experimental values of the principal stretches. The agreement of the numerical values with the experimental ones over the whole field of stretches is moder ately good in the case of order two Ogden's law and very satisfactory in the case of order three Ogden's law.

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